Block Fill
Definition For a given set of polyforms P, and a given shape S, with an area equal to the total area of P, how many ways can P be arranged to cover S. The set P The usual set used is the complete set of polyforms of a given size, such as the twelve pentominoes. Sometimes smaller sets are created by excluding polyforms that create parity problems. Or the complete set can be divided into equal subsets, and each subset used to cover the same shape. Another option can be to include all polyforms up to a specific size. The shape S The usual shapes considered are Convex Blocks, and magnifications, but just about any shape can be used. Convex Blocks For convex blocks, it is simply a matter of getting the list of blocks of the given size. Some blocks can be excluded right away, because of parity conflicts between the shape, and the set. Magnification Magnification shapes can be more difficult to find, as it will depend on the possible magnifications of the base form. For example, polyiamonds have an area of n^2 for an n x magnification, so the 72 triangles of the 12 hexaiamonds could cover 2x magnifications of the 18-iamonds or 3x magnifications of the octoiamonds. (also the 6x magnification of the diamond, but that is one of the convex blocks, and is not usually considered a magnification) This can produce an excessive number of possibilities to check, as there are 1,736,328 18-iamonds, most of which will be impossible, or very few, if the total area of P doesn't line up possible magnifications. Also, it is rare to be able to get magnifications of the same size polyforms as in P. Another common shape used is magnification with holes. Almost always the hole is a smaller version of the large shape. For example, again consider the hexaiamonds. It's 72 triangles could cover 3x magnifications of 9-iamonds with a regular 9-iamond hole, or a 4x hexaiamond with a 2x hexaiamond hole. Usually holes do not touch the edge of the magnified shape, but larger holes may not fit, otherwise. Such holes cannot divide the shape into separate pieces. Other Shapes Other shapes that can be considered are nice symmetrical shapes, or shapes that look like pictures. The more compact a shape, the more likely it is that it is possible to fill, and the more arrangements. Spindly shapes tend to have few options. Polyforms that can be used Only polyforms that can fit together tightly work. This means that poly-pents, and poly-septagons and higher cannot be used. Convex blocks There are a few forms that have convex blocks, and more that have partial convex blocks. Polyforms with convex blocks *Polyiamonds *Polyominoes *Polyetrakis *Trihexagonal tiling *Polytriakis *Polykites *Polyrhombillekis Polyform with quasi-convex blocks *Polyhex *Truncated square tiling *Possibly Rhombille tiling as it has a hex shape *Floret pentagonal tiling *Elongated triangular tiling *Prismatic pentagonal tiling Other polyforms These tilings do not have well defined straight edges in the tiling, which makes defining even quasi-convex tilings difficult. *Snub square tiling *Polycairo *Truncated hexagonal tiling *Rhombitrihexagonal tiling *Truncated trihexagonal tiling *Snub hexagonal tiling Magnifications Polyform with Magnification Polyform with quasi-magnification Other polyform Category:Polyform Topics